Klein cordial trees and odd cyclic cordial friendship graphs

TL;DR

This paper investigates the conditions under which certain graphs, especially trees and friendship graphs, admit equitable labelings based on abelian groups, revealing new classifications and conjectures in graph labeling theory.

Contribution

It proves most trees are $ ext{Z}_2^2$-cordial except for two specific paths and establishes broad conditions for friendship graphs to be $ ext{Z}_m$-cordial, proposing a new conjecture.

Findings

All trees are $ ext{Z}_2^2$-cordial except $P_4$ and $P_5$.

Friendship graphs $F_n$ are $ ext{Z}_m$-cordial when $m$ is an odd multiple of 3.

A conjecture is proposed to determine $ ext{Z}_m$-cordiality of $F_n$.

Abstract

For a graph $G$ and an abelian group $A$, a labeling of the vertices of $G$ induces a labeling of the edges via the sum of adjacent vertex labels. Hovey introduced the notion of an $A$-cordial vertex labeling when both the vertex and edge labels are as evenly distributed as possible. Much work has since been done with trees, hypertrees, paths, cycles, ladders, prisms, hypercubes, and bipartite graphs. In this paper we show that all trees are $\mathbb{Z}_2^2$-cordial except for $P_4$ and $P_5$. In addition, we give numerous results relating to $\mathbb{Z}_m$-cordiality of the friendship graph $F_n$. The most general result shows that when $m$ is an odd multiple of $3$, then $F_n$ is $\mathbb{Z}_m$-cordial for all $n$. We also give a general conjecture to determine when $F_n$ is $\mathbb{Z}_m$-cordial.